Impediments to mixing classical and quantum dynamics

Caro, J.; Salcedo, L. L.

doi:10.1103/physreva.60.842

Cited by 99 publications

(159 citation statements)

References 16 publications

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“…The numerous attempts to formulate such fundamental quantum-classical theories have encountered severe difficulties [98,99,100,101,102,103,104,105,106]. There are no-go theorems showing in which specific sense such theories cannot exist [107,108].…”

Section: Dutch Proverbmentioning

confidence: 99%

“…Following suggestions of Bohr that the proper quantum measurement should imply a classical apparatus [83,84,85], there were several attempts to work out interaction between a quantum and an explicitly classical system [95,96,97,98,99,100,101,102,103,104,105,106,107,108,109]. (Neither Bohr [83,84], nor Landau and Lifshitz [85] who present Bohr's opinion in quite detail, consider the proper interaction processes.)…”

Section: Dutch Proverbmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Dutch Proverbmentioning

confidence: 99%

Section: Dutch Proverbmentioning

confidence: 99%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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“…In Ref. [21] it is also argued that induced fluctuations in the classical sector would spoil the commutativity of the classical variables. However, this is based on assuming that the total Hamiltonian is bounded from below; as we will see, in the Koopman-von Neumann-Sudarshan approach this boundedness is lost [10], so that one can accommodate commuting variables with induced fluctuations.…”

Section: Introductionmentioning

confidence: 99%

“…Approaches of type 1 have been shown to lead to a number of inconsistencies [21]. On the one hand, the standard semiclassical treatment [2] does not transfer the primary fluctuations in the quantum sector into the classical sector.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, the dynamics is dictated by a bracket failing to satisfy Jacobi's identity and Leibniz's rule. Under reasonable hypotheses it has been shown that there is no Lie bracket for a semiquantized theory of this form [21,[25][26][27][28]. If one permits the bracket to be of non-Lie type, it can be shown that the theory will lose the positivity of its density matrix [2,21] (positive operators could have nonpositive expectation values).…”

Section: Introductionmentioning

confidence: 99%

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Hybrid classical-quantum formulations ask for hybrid notions

et al. 2012

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We reappraise some of the hybrid classical-quantum models proposed in the literature with the goal of retrieving some of their common characteristics. In particular, first, we analyze in detail the Peres-Terno argument regarding the inconsistency of hybrid quantizations of the Sudarshan type. We show that to accept such hybrid formalism entails the necessity of dealing with additional degrees of freedom beyond those in the straight complete quantization of the system. Second, we recover a similar enlargement of degrees of freedom in the so-called statistical hybrid models. Finally, we use Wigner's quantization of a simple model to illustrate how in hybrid systems the subsystems are never purely classical or quantum. A certain degree of quantumness (classicality) is being exchanged between the different sectors of the theory, which in this particular unphysical toy model makes them undistinguishable.

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Dynamics of Ground‐ and Excited‐State Intramolecular Proton Transfer Reactions

Zwan

2013

Tautomerism

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Impediments to mixing classical and quantum dynamics

Cited by 99 publications

References 16 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

Hybrid classical-quantum formulations ask for hybrid notions

Dynamics of Ground‐ and Excited‐State Intramolecular Proton Transfer Reactions

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